Eigenvectors, Eigenvalues and Idempotent

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An n x n matrix A is idempotent if A^2 = A, which leads to the conclusion that its eigenvalues must be either 0 or 1. To demonstrate this, one must consider a nonzero vector v such that A*v = λ*v, and then apply A again to manipulate the equation. By rearranging the terms, it becomes possible to isolate λ and derive its possible values. This approach effectively shows the relationship between idempotent matrices and their eigenvalues. Understanding this concept is crucial for further studies in linear algebra.
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I have a question that deals with all three of the terms in the title. I'm not really even sure where to begin on this. I was hoping someone could help.

Question:

An n x n matrix A is said to be idempotent if A^2 = A. Show that if λ is an eigenvalue of an independent matrix, then λ must either be 0 or 1.

If I could get some help on this I would really appreciate it.

Thanks,

mpm
 
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Well you know that there must exist a nonzero vector v such that A*v=lamda*v. Now play with this statement by applying A again, and rearanging terms so that you end up with only expressions involving lambda. Then solve for lambda.
 

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